This paper studies the existence of solutions for Caputo-Hadamard fractional nonlinear differential equations of variable order (CHFDEVO). We obtain some needed conditions for this purpose by providing an auxiliary constant order system of the given CHFDEVO. In other words, with the help of piece-wise constant order functions on some continuous subintervals of a partition, we convert the main variable order initial value problem (IVP) to a constant order IVP of the Caputo-Hadamard differential equations. By calculating and obtaining equivalent solutions in the form of a Hadamard integral equation, our results are established with the help of the upper-lower-solutions method. Finally, a numerical example is presented to express the validity of our results.
Citation: Zoubida Bouazza, Sabit Souhila, Sina Etemad, Mohammed Said Souid, Ali Akgül, Shahram Rezapour, Manuel De la Sen. On the Caputo-Hadamard fractional IVP with variable order using the upper-lower solutions technique[J]. AIMS Mathematics, 2023, 8(3): 5484-5501. doi: 10.3934/math.2023276
[1] | Alaa Altassan, Muhammad Imran, Bilal Ahmad Rather . On ABC energy and its application to anticancer drugs. AIMS Mathematics, 2023, 8(9): 21668-21682. doi: 10.3934/math.20231105 |
[2] | Gulalai, Shabir Ahmad, Fathalla Ali Rihan, Aman Ullah, Qasem M. Al-Mdallal, Ali Akgül . Nonlinear analysis of a nonlinear modified KdV equation under Atangana Baleanu Caputo derivative. AIMS Mathematics, 2022, 7(5): 7847-7865. doi: 10.3934/math.2022439 |
[3] | Bahar Acay, Ramazan Ozarslan, Erdal Bas . Fractional physical models based on falling body problem. AIMS Mathematics, 2020, 5(3): 2608-2628. doi: 10.3934/math.2020170 |
[4] | Ramazan Ozarslan, Erdal Bas, Dumitru Baleanu, Bahar Acay . Fractional physical problems including wind-influenced projectile motion with Mittag-Leffler kernel. AIMS Mathematics, 2020, 5(1): 467-481. doi: 10.3934/math.2020031 |
[5] | Wanxia Wang, Shilin Yang . On finite-dimensional irreducible modules for the universal Askey-Wilson algebra. AIMS Mathematics, 2023, 8(8): 18930-18947. doi: 10.3934/math.2023964 |
[6] | Saima Rashid, Fahd Jarad, Fatimah S. Bayones . On new computations of the fractional epidemic childhood disease model pertaining to the generalized fractional derivative with nonsingular kernel. AIMS Mathematics, 2022, 7(3): 4552-4573. doi: 10.3934/math.2022254 |
[7] | Saeed M. Ali, Mohammed S. Abdo, Bhausaheb Sontakke, Kamal Shah, Thabet Abdeljawad . New results on a coupled system for second-order pantograph equations with ABC fractional derivatives. AIMS Mathematics, 2022, 7(10): 19520-19538. doi: 10.3934/math.20221071 |
[8] | Saima Rashid, Fahd Jarad, Taghreed M. Jawa . A study of behaviour for fractional order diabetes model via the nonsingular kernel. AIMS Mathematics, 2022, 7(4): 5072-5092. doi: 10.3934/math.2022282 |
[9] | Rahat Zarin, Abdur Raouf, Amir Khan, Aeshah A. Raezah, Usa Wannasingha Humphries . Computational modeling of financial crime population dynamics under different fractional operators. AIMS Mathematics, 2023, 8(9): 20755-20789. doi: 10.3934/math.20231058 |
[10] | Ahu Ercan . Comparative analysis for fractional nonlinear Sturm-Liouville equations with singular and non-singular kernels. AIMS Mathematics, 2022, 7(7): 13325-13343. doi: 10.3934/math.2022736 |
This paper studies the existence of solutions for Caputo-Hadamard fractional nonlinear differential equations of variable order (CHFDEVO). We obtain some needed conditions for this purpose by providing an auxiliary constant order system of the given CHFDEVO. In other words, with the help of piece-wise constant order functions on some continuous subintervals of a partition, we convert the main variable order initial value problem (IVP) to a constant order IVP of the Caputo-Hadamard differential equations. By calculating and obtaining equivalent solutions in the form of a Hadamard integral equation, our results are established with the help of the upper-lower-solutions method. Finally, a numerical example is presented to express the validity of our results.
Let G be a simple connected graph with vertex set V(G)={v1,v2,…,vn} and edge set E(G). The eigenvalues of adjacency matrix A(G) are called the eigenvalues of G. The energy E(G) of G is defined as the sum of the absolute values of its eigenvalues of A(G), which is studied in chemistry and used to approximate the total-electron energy of a molecule [3]. The singular values of an n×m matrix M are the square roots of the eigenvalues of MM∗ if n≥m or M∗M if n<m, where M∗ is the transpose conjugate of M. Nikiforov [4] extended the concept of energy to all matrices and defined the energy of a matrix M, denoted by E(M), as the sum of the singular values of M. Clearly, E(A(G))=E(G).
Estrada et al. [12] introduced the atom-bond connectivity index as
ABC(G)=∑vivj∈E(G)√di+dj−2didj. |
Moreover, they introduced the atom-bond connectivity matrix (or ABC matrix for short) ABCG of G, which is correlated with the ABC index of G. The (i,j)-entry of the matrix ABCG is equal to √di+dj−2didj if vivj∈E(G) and 0 otherwise. The eigenvalues of the ABC matrix of G, denoted by μ1,μ2,…,μn, are said to be the ABC eigenvalues of G. The atom-bond connectivity energy (ABC energy) of a connected graph G is defined in [8] as
EABC(G)=n∑i=1|μi(G)|. |
Recently, several theoretical and computational properties of the ABC energy of graphs have been obtained, see e.g., [1,8,13]. Estrada [8] and Chen [13] gave an upper bound and a lower bound for the ABC energy in terms of the general Randić index, respectively. Ghorbani et al. [1] established some new bounds for the ABC energy. Gao and Shao [7] determined the unique tree with the minimum ABC energy. In this paper, we determine the trees with the minimum ABC energy among all trees on n vertices except the star Sn.
A matching in a graph is a set of edges without common vertices. A k-matching is a matching consisting of k edges. Let T be a tree, M be a matching of T and Mk(T) be the set of all k-matchings of T. We define m∗M(T) and m∗(T,k) by
m∗M(T)=∏vivj∈M(ABCT)2ij |
and
m∗(T,k)=∑M∈Mk(T)m∗M(T), |
respectively. By Sachs Theorem [14], the characteristic polynomial ϕABC(T,x) of the ABC matrix of a tree T can be expressed as
ϕABC(T,x)=⌊n2⌋∑k=0(−1)km∗(T,k)xn−2k. |
Then by Coulson integral formula, we get
EABC(T)=2π∫+∞01x2ln[1+⌊n2⌋∑k=1m∗(T,k)x2k]dx. | (2.1) |
Let T1 and T2 be two trees on n vertices. If m∗(T1,k)≥m∗(T2,k) for all k, then by (2.1) we get EABC(T1)≥EABC(T2). Moreover, if there exists some k such that m∗(T1,k)>m∗(T2,k), then EABC(T1)>EABC(T2).
Let T be a tree on n vertices, B=(bij) be an n×n nonnegative real symmetric matrix and ABCT≥B. Let M be a matching of T, m∗M(B)=∏vivj∈Mb2ij and m∗(B,k)=∑M∈Mk(T)m∗M(B). Then
E(B)=2π∫+∞01x2ln[1+⌊n2⌋∑k=1m∗(B,k)x2k]dx. |
Clearly, m∗(T,k)≥m∗(B,k). Thus EABC(T)≥E(B). Moreover, if (ABCT)ij>bij for some vivj∈E(T), then EABC(T)>E(B). Thus we can get the following lemma.
Lemma 2.1. Let T be a tree on n vertices and B be an n×n nonnegative real symmetric matrix. If ABCT≥B, then EABC(T)≥E(B). Moreover, if (ABCT)ij>bij for some vivj∈E(T), then EABC(T)>E(B).
Let uv be an edge of a tree T and T−uv=T1∪T2, where T1(T2) is the component of T−uv containing u(v, respectively). We denote the sub-matrices of ABCT spanned by the vertices of T1 and T2 by (ABCT)V(T1) and (ABCT)V(T2), respectively.
By Lemma 2.1, we have the next lemma.
Lemma 2.2. Let uv be an edge of a tree T and T−uv=T1∪T2, where T1(T2) is the component of T−uv containing u(v, respectively). Then
EABC(T)>E((ABCT)V(T1))+E((ABCT)V(T2)). |
Suppose that uv is not a pendent edge. If dT(w)≤2 for any w∈NT(u)∖{v}, then
E((ABCT)V(T1))≥EABC(T1). |
Furthermore, if d(w)≤2 for any w∈NT(u)∪NT(v)∖{u,v}, then
EABC(T)>EABC(T1)+EABC(T2). |
Lemma 2.3. ([7]) Let T be a tree of order n≥3. Then EABC(T)≥2√n−2, withequality if and only if T≅Sn, where Sn is the star of order n.
Lemma 2.4. ([7]) Let t≥2,xi≥3 for i=1,…,t, and ∑ti=1xi≥8. Then ∑ti=2√xi−2≥√∑ti=2xi+(t−1)−2.
For two graphs G and H, we define G∪H to be their disjoint union. In addition, let kG be the disjoint union of k copies of G. Let S∗n be the tree formed by attaching a vertex to a pendent vertex of the star Sn−1. Note that
ϕABC(S∗n,x)=xn−4[x4−(1+(n−3)2n−2)x2+(n−3)22(n−2)]. |
Thus
EABC(S∗n)=2√n−3+1n−2+√2√n−4+1n−2. |
Lemma 3.1. Let x≥11. Then
√x−5+1>√x−3+1x−2+√2√x−4+1x−2. | (3.1) |
Proof. It is equivalent to prove that
2√x−5−√2√x−4+1x−2−1x−2−1>0. | (3.2) |
Let f(x)=2√x−5−√2√x−4+1x−2−1x−2−1 with x≥11. Then
dfdx=1√x−5−√221√x−4+1x−2(1−1(x−2)2)+1(x−2)2>1√x−5−√221√x−4+1x−2=1√x−5−1√2(x−4)+2x−2>0. |
Thus f(x) is a strictly monotonously increasing function on x. Noting that f(11)=0.0166>0, then the lemma holds.
From Lemma 3.1, EABC(S∗n)<2+2√n−5 for n≥11.
For n=1,2,3, there is only unique tree Sn. For n=4, there are exactly two trees P4 and S4. Obviously, P4 is the tree with the second minimum ABC energy. For n=5, there are exactly three trees P5, S5 and S∗5. By direct calculation, we have EABC(S∗5)=3.9831>EABC(P5)=√2+√6>2√3=EABC(S5). Thus P5 is the tree with the second minimum ABC energy. Let P5=v1v2v3v4v5, we denote the tree, obtained by attaching a new vertex to v2 of P5, by P∗6. For n=6, there are exactly six trees T2.8,T2.9,T2.10,T2.11,T2.12,T2.13 (see tables of graph spectra in [14]), where T2.8≅S6,T2.9≅S∗6, T2.11≅P∗6 and T2.13≅P6. By direct calculation, EABC(T2.12)=5.0590>EABC(P6)=4.9412>EABC(T2.10)=4.8074>EABC(S∗6)=4.6352>EABC(P∗6)=4.6260>EABC(S6)=4.
By simple calculations, we obtain the following lemma.
Lemma 3.2. Let T be an n-vertex tree not isomorphic to Sn, where 7≤n≤10. Then EABC(T)≥EABC(S∗n) with equality if and only if T≅S∗n.
Lemma 3.3. Let T be a tree on n≥11 vertices.
(i) Let u1v1∈E(G) and T−u1v1=T1∪T2, where T1(T2) is the component of T−u1v1 containing u1(v1, respectively). If d(w)≤2 for any w∈N(u1)∪N(v1)∖{u1,v1} and |V(T1)|=n1≥|V(T2)|=n2≥3, then EABC(T)>EABC(S∗n).
(ii) Let u2v2,u3v3∈E(G), T−u2v2≅P2∪T3 and T3−u3v3≅P2∪T4, where T3 is one of the component of T−u2v2 and T4 is one of the component of T3−u3v3. If dT(w1)≤2 for any w1∈NT(u2)∪NT(v2)∖{u2,v2} and dT3(w2)≤2 for any w2∈NT3(u3)∪NT3(v3)∖{u3,v3}, then EABC(T)>EABC(S∗n).
Proof. (ⅰ) By Lemmas 2.2 and 2.3, we have
EABC(T)>EABC(T1)+EABC(T2)≥2√n1−2+2√n2−2≥2√n−5+2>EABC(S∗n). |
(ⅱ) Similarly, by Lemmas 2.2 and 2.3, we have
EABC(T)>EABC(T3)+E((ABCT)V(P2))>EABC(T4)+√2+√2≥2√n−6+2√2≥2√n−5+2>EABC(S∗n). |
We complete the proof.
Lemma 3.4. Let n≥11. Then EABC(Pn)>EABC(S∗n).
Proof. By Lemma 2.2, we have EABC(Pn)>EABC(P3)+EABC(Pn−3)≥2+2√n−5>EABC(S∗n).
A tree is called starlike if it has exactly one vertex of degree greater than two.
Lemma 3.5. Let T≇Sn be a starlike tree with order n≥11 and v be the unique vertex with degree at least three. Let T−v=n1P1∪n2P2∪⋯∪nmPm and ∑mi=1ini+1=n. Then EABC(T)>EABC(S∗n).
Proof. If ni≥1 for some i≥3, then there exists an edge uv such that T−uv=T1∪T2, where T1(T2) is the component of T−uv containing u(v, respectively), |V(T1)|,|V(T2)|≥3 and d(w)≤2 for any w∈N(u)∪N(v)∖{u,v}. Then by (ⅰ) of Lemma 3.3 we can get the result. Suppose that ni=0 for all i≥3. If n2=0, then T≅Sn. If n2=1, then T≅S∗n. If n2≥2, then by (ⅱ) of Lemma 3.3 we can get the result.
Let T be a tree and R(T) be set of vertices of degree greater than two in T.
Lemma 3.6. Let T be a tree with n≥11 vertices and |R(T)|≥2. If there are no adjacent vertices in R(T), then EABC(T)>EABC(S∗n).
Proof. Let d(u)≥3 and d(v)≥3 and Pl=uv1…vl−1v be the single path connecting u and v with d(v1)=⋯=d(vl−1)=2. Clearly, l≥2. Without loss of generality, we suppose that T−uv1=T1∪T2 such that T1 is a starlike-tree or a path, where u∈V(T1).
If l≥3, then by (i) of Lemma 3.3 we can get the result.
Suppose now that l=2. Let T′1(T′2) be the component of T−uv1−v1v containing u(v, respectively), s=|V(T′1)| and t=|V(T′2)|. Obviously, s+t+1=n.
If s=3, then the ABC matrix of T can be written as
ABCT=[BCC⊤D], |
where
B=[00√23000√230√23√230√1200√120],C=[03×103×(t−1)√1201×(t−1)], |
and D=(ABCT)V(T′2). Let
A=[B00ABCT′2]. |
Obviously, D≥ABCT′2. Thus ABCT>A. By Lemmas 2.1 and 2.3, we have
EABC(T)>E(A)=E(B)+EABC(T′2)≥2√1+13+12+2√n−6≥2√n−5+2>EABC(S∗n). |
Suppose that s=4. Then T′1≅S4 or P4. If T′1≅S4, then the ABC matrix of T can be written as
ABCT=[FHH⊤K], |
where
F=[000√340000√340000√340√34√34√340√12000√120],H=[04×104×(t−1)√1201×(t−1)], |
and K=(ABCT)V(T′2). Let
M=[F00ABCT′2]. |
Obviously, K≥ABCT′2. Thus ABCT>M. By Lemmas 2.1 and 2.3 we have
EABC(T)≥E(M)=E(F)+EABC(T′2)≥2√2+14+12+2√n−7≥2√n−5+2>EABC(S∗n). |
Suppose now that T′1≅P4. Let T′1=u1uu2u3. Then by Lemmas 2.2 and 2.3, we have
EABC(T)≥√2+EABC(T−u2−u3)≥√2+EABC(P3)+EABC(T′2)≥√2+2+2√n−7≥2√n−5+2>EABC(S∗n). |
By symmetry, we now suppose that 5≤s,t≤n−6, then by Lemmas 2.2 and 2.3, we have
EABC(T)>EABC(T′1)+EABC(T′2)≥2√s−2+2√t−2≥2√3+2√n−8>2+2√n−5>EABC(S∗n). |
Lemma 3.7. Let T be a tree with n≥11 vertices and |R(T)|≥2. If there exist adjacent vertices in R(T), then EABC(T)>EABC(S∗n).
Proof. Let E0={uv∈E(T)|d(u),d(v)≥3}, and T−E0=xP1∪yP2∪T1∪⋯∪Tz, where T1,…,Tz are components of T−E0 with at least three vertices. Let xP1={v1,…,vx} and yP2={vx+1vx+2,…,vx+2y−1vx+2y}. Then dT(vi)≥3 with 1≤i≤x, dT(vx+2j−1)≥3 and dT(vx+2j)=1 with 1≤j≤y, and for each component Ti with 1≤i≤z, there exists a vertex vi∈V(Ti) such that dT(vi)≥dTi(vi)+1. Let |V(Ti)|=si for 1≤i≤z. Thus we have
2(n−1)=∑v∈V(T)dT(v)≥3x+4y+z∑i=1(∑v∈V(Ti)dTi(v)+1)=3x+4y+z∑i=12(si−1)+z=x+2n−z. |
Thus we get that z≥x+2. We discuss the following four cases.
Case 1. y=0 and z=2.
Then x=0 and s1+s2=n. By Lemmas 2.2 and 2.3, we get
EABC(T)>EABC(T1)+EABC(T2)≥2√s1−2+2√s2−2≥2√n−5+2>EABC(S∗n). |
Case 2. y=0 and z≥3.
Then x+∑zi=1si=n. Without loss of generation, we suppose that 3≤sz≤sz−1≤⋯≤s2≤s1.
If ∑z−1i=1si=6, then z=3,sz=3 and x≤1. Thus n≤10, a contradiction.
If ∑z−1i=1si=7, then z=3,s1=4,s2=s3=3. Thus x=1 and n=11. Obviously, T2≅T3≅S3 and T1≅S4 or P4. By Lemmas 2.2 and 2.3, we have
EABC(T)≥EABC(T1)+E((ABCT)V(T2))+E((ABCT)V(T3))≥2√4−2+4×2√3=7.448>6.8742=EABC(S∗11). |
Suppose that ∑z−1i=1si≥8. By Lemmas 2.2–2.4, we have that
EABC(T)>z∑i=1EABC(Ti)≥2z∑i=1√si−2≥2√z−1∑i=1si+(z−1)−3+2√sz−2≥2√n−x−sz+x+2−4+2√sz−2=2√n−sz−2+2√sz−2≥2√n−5+2>EABC(S∗n). | (3.3) |
Case 3. y≥1 and z≥3.
Then ∑z−1i=1si+sz+x+2y=n. By Lemmas 2.2 and 2.3, we have
EABC(T)≥2y√23+z∑i=1EABC(Ti)≥2y√23+2z∑i=1√si−2. |
If ∑z−1i=1si≥8, then by Lemma 2.4, we have
EABC(T)≥2y√23+2√z−1∑i=1si+(z−1−1)−2+2√sz−2≥2y√23+2√n−sz−2y−x+x+2−4+2√sz−2=2y√23+2√n−sz−2y−2+2√sz−2≥2y√23+2√n−3−2y−2+2√3−2≥2√n−5+2>EABC(S∗n). |
Here, the last but one inequality holds because f(y)=2y√23+2√n−5−2y+2 is increasing for 0≤y≤2n−134.
Suppose that ∑z−1i=1si≤7. Then z=3. Thus s1+s2=6 or 7, s3=3 and x≤1.
Suppose first that s1+s2=7 and s3=3. If x=0, then n=2y+10. Hence
EABC(T)≥2y√23+EABC(T1)+EABC(T2)+EABC(T3)≥2y√23+2√2+2+2=(n−10)√23+2√2+4≥2√n−5+2>EABC(S∗n). |
Suppose now that x=1. Then n=2y+11. Hence
EABC(T)>2y√23+2√2+2+2=(n−11)√23+2√2+4. |
Let f(x)=(x−11)√23+4+2√2−2√x−3+1x−2+√2⋅√x−4+1x−2. It is easy to get that f′(x)>0 for x≥11. Then f(x) is an increasing function on x and f(x)≥f(11)>0. Thus
EABC(T)>(n−11)√23+4+2√2>2√n−3+1n−2+√2⋅√n−4+1n−2. |
By a similar discussion as above, we can get the result for the case s1+s2=6 and s3=3.
Case 4. y≥1 and z=2.
Then x=0,n=2y+s1+s2. If n−2y≥11, then by Lemmas 2.1–2.3, we have
EABC(T)≥2y√23+EABC(T1)+EABC(T2)≥2y√23+2√s1−2+2√s2−2≥2y√23+2√n−2y−3−2+2√3−2≥2√n−5+2>EABC(S∗n). |
Suppose that n−2y≤10. Then 6≤s1+s2≤10. If s1+s2=10, then by Lemmas 2.1–2.3, we have
EABC(T)>2y√23+2√3−2+2√7−2=(n−10)√23+2+2√5≥2√n−5+2>EABC(S∗n). |
Similarly, for each 6≤s1+s2≤9, we may also get the result.
Combining Lemmas 3.2 and 3.4–3.7, we get our main result.
Theorem 3.1. Among all trees (except the star) on n≥5 vertices, P5 is the unique tree with the minimum ABC energy for n=5, P∗6 is the unique tree with the minimum ABC energy for n=6 and S∗n is the unique tree with the minimum ABC energy for n≥7.
In this paper, motivated by the unique tree with the minimum ABC energy, we determine the trees with the minimum ABC energy among all trees on n vertices except the star Sn.
The authors would like to thank anonymous referees for helpful comments and suggestions which improved the original version of the paper. This work was supported by the Natural Science Foundation of Guangdong Province (No.2021A1515010028).
The authors declare that they have no competing interests.
[1] |
A. O. Akdemir, A. Karaoǧlan, M. A. Ragusa, E. Set, Fractional integral inequalities via Atangana-Baleanu operators for convex and concave functions, J. Funct. Spaces, 2021 (2021), 1055434. https://doi.org/10.1155/2021/1055434 doi: 10.1155/2021/1055434
![]() |
[2] |
M. S. Abdo, Further results on the existence of solutions for generalized fractional quadratic functional integral equations, J. Math. Anal. Model., 1 (2020), 33–46. https://doi.org/10.48185/jmam.v1i1.2 doi: 10.48185/jmam.v1i1.2
![]() |
[3] |
R. Rizwan, A. Zada, X. Wang, Stability analysis of nonlinear implicit fractional Langevin equation with noninstantaneous impulses, Adv. Differ. Equ., 2019 (2019), 85. https://doi.org/10.1186/s13662-019-1955-1 doi: 10.1186/s13662-019-1955-1
![]() |
[4] |
D. Baleanu, S. Etemad, S. Rezapour, A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions, Bound. Value Probl., 2020 (2020), 64. https://doi.org/10.1186/s13661-020-01361-0 doi: 10.1186/s13661-020-01361-0
![]() |
[5] |
A. Zada, J. Alzabut, H. Waheed, I. L. Popa, Ulam-Hyers stability of impulsive integrodifferential equations with Riemann-Liouville boundary conditions, Adv. Differ. Equ., 2020 (2020), 64. https://doi.org/10.1186/s13662-020-2534-1 doi: 10.1186/s13662-020-2534-1
![]() |
[6] |
E. Bonyah, C. W. Chukwu, M. L. Juga, Fatmawati, Modeling fractional-order dynamics of Syphilis via Mittag-Leffler law, AIMS Math., 6 (2021), 8367–8389. https://doi.org/10.3934/math.2021485 doi: 10.3934/math.2021485
![]() |
[7] |
M. S. Abdo, T. Abdeljawad, S. M. Ali, K. Shah, F. Jarad, Existence of positive solutions for weighted fractional order differential equations, Chaos Solitons Fract., 141 (2020), 110341. https://doi.org/10.1016/j.chaos.2020.110341 doi: 10.1016/j.chaos.2020.110341
![]() |
[8] |
A. Atangana, S. İ. Araz, Nonlinear equations with global differential and integral operators: existence, uniqueness with application to epidemiology, Results Phys., 20 (2021), 103593. https://doi.org/10.1016/j.rinp.2020.103593 doi: 10.1016/j.rinp.2020.103593
![]() |
[9] |
H. Mohammad, S. Kumar, S. Rezapour, S. Etemad, A theoretical study of the Caputo-Fabrizio fractional modeling for hearing loss due to Mumps virus with optimal control, Chaos Solitons Fract., 144 (2021), 110668. https://doi.org/10.1016/j.chaos.2021.110668 doi: 10.1016/j.chaos.2021.110668
![]() |
[10] |
S. Etemad, I. Iqbal, M. E. Samei, S. Rezapour, J. Alzabut, W. Sudsutad, et al., Some inequalities on multi-functions for applying in the fractional Caputo-Hadamard jerk inclusion system, J. Inequal. Appl., 2022 (2022), 84. https://doi.org/10.1186/s13660-022-02819-8 doi: 10.1186/s13660-022-02819-8
![]() |
[11] |
H. Khan, K. Alam, H. Gulzar, S. Etemad, S. Rezapour, A case study of fractal-fractional tuberculosis model in China: existence and stability theories along with numerical simulations, Math. Comput. Simul., 198 (2022), 455–473. https://doi.org/10.1016/j.matcom.2022.03.009 doi: 10.1016/j.matcom.2022.03.009
![]() |
[12] |
S. Belmor, F. Jarad, T. Abdeljawad, G. Kınıç, A study of boundary value problem for generalized fractional differential inclusion via endpoint theory for weak contractions, Adv. Differ. Equ., 2020 (2020), 348. https://doi.org/10.1186/s13662-020-02811-w doi: 10.1186/s13662-020-02811-w
![]() |
[13] |
S. Rezapour, M. I. Abbas, S. Etemad, N. M. Dien, On a multipoint p-Laplacian fractional differential equation with generalized fractional derivatives, Math. Meth. Appl. Sci., 2022. https://doi.org/10.1002/mma.8301 doi: 10.1002/mma.8301
![]() |
[14] |
A. M. Saeed, M. S. Abdo, M. B. Jeelani, Existence and Ulam-Hyers stability of a fractional order coupled system in the frame of generalized Hilfer derivatives, Mathematics, 9 (2021), 2543. https://doi.org/10.3390/math9202543 doi: 10.3390/math9202543
![]() |
[15] |
S. Etemad, I. Avci, P. Kumar, D. Baleanu, S. Rezapour, Some novel mathematical analysis on the fractal-fractional model of the AH1N1/09 virus and its generalized Caputo-type version, Chaos Solitons Fract., 162 (2022), 112511. https://doi.org/10.1016/j.chaos.2022.112511 doi: 10.1016/j.chaos.2022.112511
![]() |
[16] |
J. F. Gómez-Aguilar, Analytical and numerical solutions of nonlinear alcoholism model via variable-order fractional differential equations, Phys. A: Stat. Mech. Appl., 494 (2018), 52–75. https://doi.org/10.1016/j.physa.2017.12.007 doi: 10.1016/j.physa.2017.12.007
![]() |
[17] |
H. G. Sun, W. Chen, H. Wei, Y. Q. Chen, A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems, Eur. Phys. J. Spec. Top., 193 (2011), 185–192. https://doi.org/10.1140/epjst/e2011-01390-6 doi: 10.1140/epjst/e2011-01390-6
![]() |
[18] |
D. Tavares, R. Almeida, D. F. M. Torres, Caputo derivatives of fractional variable order Numerical approximations, Commun. Nonlinear Sci. Numer. Simul., 35 (2016), 69–87. https://doi.org/10.1016/j.cnsns.2015.10.027 doi: 10.1016/j.cnsns.2015.10.027
![]() |
[19] |
J. V. da C. Sousa, E. C. de Oliverira, Two new fractional derivatives of variable order with non-singular kernal and fractional differential equation, Comp. Appl. Math., 37 (2018), 5375–5394. https://doi.org/10.1007/s40314-018-0639-x doi: 10.1007/s40314-018-0639-x
![]() |
[20] |
J. Yang, H. Yao, B. Wu, An efficient numberical method for variable order fractional functional differential equation, Appl. Math. Lett., 76 (2018), 221–226. https://doi.org/10.1016/j.aml.2017.08.020 doi: 10.1016/j.aml.2017.08.020
![]() |
[21] | J. H. An, P. Y. Chen, P. Chen, Uniqueness of solutions to initial value problem of fractional differential equations of variable-order, Dyn. Syst. Appl., 28 (2019), 607–623. |
[22] |
Z. Bouazza, S. Etemad, M. S. Souid, S. Rezapour, F. Martínez, M. K. A. Kaabar, A study on the solutions of a multiterm FBVP of variable order, J. Funct. Spaces, 2021 (2021), 9939147. https://doi.org/10.1155/2021/9939147 doi: 10.1155/2021/9939147
![]() |
[23] |
A. Benkerrouche, M. S. Souid, K. Sitthithakerngkiet, A. Hakem, Implicit nonlinear fractional differential equations of variable order, Bound. Value Probl., 2021 (2021), 64. https://doi.org/10.1186/s13661-021-01540-7 doi: 10.1186/s13661-021-01540-7
![]() |
[24] |
A. Refice, M. S. Souid, I. Stamova, On the boundary value problems of Hadamard fractional differential equations of variable order via Kuratowski MNC technique, Mathematics, 9 (2021), 1134. https://doi.org/10.3390/math9101134 doi: 10.3390/math9101134
![]() |
[25] |
S. Hristova, A. Benkerrouche, M. S. Souid, A. Hakem, Boundary value problems of Hadamard fractional differential equations of variable order, Symmetry, 13 (2021), 896. https://doi.org/10.3390/sym13050896 doi: 10.3390/sym13050896
![]() |
[26] |
S. G. Samko, B. Ross, Integration and differentiation to a variable fractional order, Integr. Trans. Spec. F., 1 (1993), 277–300. https://doi.org/10.1080/10652469308819027 doi: 10.1080/10652469308819027
![]() |
[27] |
S. Zhang, S. Li, L. Hu, The existeness and uniqueness result of solutions to initial value problems of nonlinear diffusion equations involving with the conformable variable derivative, RACSAM, 113 (2019), 1601–1623. https://doi.org/10.1007/s13398-018-0572-2 doi: 10.1007/s13398-018-0572-2
![]() |
[28] |
S. Rezapour, M. S. Souid, Z. Bouazza, A. Hussain, S. Etemad, On the fractional variable order thermostat model: existence theory on cones via piece-wise constant functions, J. Funct. Spaces, 2022 (2022), 8053620. https://doi.org/10.1155/2022/8053620 doi: 10.1155/2022/8053620
![]() |
[29] |
S. Rezapour, Z. Bouazza, M. S. Souid, S. Etemad, M. K. A. Kaabar, Darbo fixed point criterion on solutions of a Hadamard nonlinear variable order problem and Ulam-Hyers-Rassias stability, J. Funct. Spaces, 2022 (2022), 1769359. https://doi.org/10.1155/2022/1769359 doi: 10.1155/2022/1769359
![]() |
[30] |
A. Ben Makhlouf, A novel finite time stability analysis of nonlinear fractional-order time delay systems: a fixed point approach, Asian J. Control, 24 (2022), 3580–3587. https://doi.org/10.1002/asjc.2756 doi: 10.1002/asjc.2756
![]() |
[31] |
A. Ben Makhlouf, Partial practical stability for fractional‐order nonlinear systems, Math. Meth. Appl. Sci., 45 (2022), 5135–5148. https://doi.org/10.1002/mma.8097 doi: 10.1002/mma.8097
![]() |
[32] |
A. Ben Makhlouf, D. Baleanu, Finite time stability of fractional order systems of neutral type, Fractal Fract., 6 (2022), 289. https://doi.org/10.3390/fractalfract6060289 doi: 10.3390/fractalfract6060289
![]() |
[33] |
H. Arfaoui, A. Ben Makhlouf, Stability of a time fractional advection-diffusion system, Chaos, Solitons Fract., 157 (2022), 111949. https://doi.org/10.1016/j.chaos.2022.111949 doi: 10.1016/j.chaos.2022.111949
![]() |
[34] |
R. Almeida, Caputo-Hadamard fractional derivatives of variable order, Numer. Funct. Anal. Opt., 38 (2017), 1–19. https://doi.org/10.1080/01630563.2016.1217880 doi: 10.1080/01630563.2016.1217880
![]() |
[35] |
A. Ben Makhlouf, L. Mchiri, Some results on the study of Caputo-Hadamard fractional stochastic differential equations, Chaos Solitons Fract., 155 (2022), 111757. https://doi.org/10.1016/j.chaos.2021.111757 doi: 10.1016/j.chaos.2021.111757
![]() |
[36] |
K. Abuasbeh, R. Shafqat, A. U. K. Niazi, M. Awadalla, Nonlocal fuzzy fractional stochastic evolution equations with fractional Brownian motion of order (1,2), AIMS Math., 7 (2022), 19344–19358. https://doi.org/10.3934/math.20221062 doi: 10.3934/math.20221062
![]() |
[37] |
K. Abuasbeh, R. Shafqat, A. U. K. Niazi, M. Awadalla, Local and global existence and uniqueness of solution for class of fuzzy fractional functional evolution equation, J. Funct. Spaces, 2022 (2022), 7512754. https://doi.org/10.1155/2022/7512754 doi: 10.1155/2022/7512754
![]() |
[38] |
A. Khan, R. Shafqat, A. U. K. Niazi, Existence results of fuzzy delay impulsive fractional differential equation by fixed point theory approach, J. Funct. Spaces, 2022 (2022), 4123949. https://doi.org/10.1155/2022/4123949 doi: 10.1155/2022/4123949
![]() |
[39] |
K. Abuasbeh, R. Shafqat, A. U. K. Niazi, M. Awadalla, Local and global existence and uniqueness of solution for time-fractional fuzzy Navier-Stokes equations, Fractal Fract., 6 (2022), 330. https://doi.org/10.3390/fractalfract6060330 doi: 10.3390/fractalfract6060330
![]() |
[40] |
R. Shafqat, A. U. K. Niazi, M. Yavuz, M. B. Jeelani, K. Saleem, Mild solution for the time-fractional Navier-Stokes equation incorporating MHD effects, Fractal Fract., 6 (2022), 580. https://doi.org/10.3390/fractalfract6100580 doi: 10.3390/fractalfract6100580
![]() |
[41] |
R. Shafqat, A. U. K. Niazi, M. B. Jeelani, N. H. Alharthi, Existence and uniqueness of mild solution where α∈(1,2) for fuzzy fractional evolution equations with uncertainty, Fractal Fract., 6 (2022), 65. https://doi.org/10.3390/fractalfract6020065 doi: 10.3390/fractalfract6020065
![]() |
[42] |
S. N. Rao, A. H. Msmali, M. Singh, A. Ali, H. Ahmadini, Existence and uniqueness for a system of Caputo-Hadamard fractional differential equations with multipoint boundary conditions, J. Funct. Spaces, 2020 (2020), 8821471. https://doi.org/10.1155/2020/8821471 doi: 10.1155/2020/8821471
![]() |
[43] |
C. Derbazi, H. Hammouche, Caputo-Hadamard fractional differential equations with nonlocal fractional integro-differential boundary conditions via topological degree theory, AIMS Math., 5 (2020), 2694–2709. https://doi.org/10.3934/math.2020174 doi: 10.3934/math.2020174
![]() |
[44] |
M. Gohar, C. Li, Z. Li, Finite difference methods for Caputo-Hadamard fractional differential equations, Mediterr. J. Math., 17 (2020), 194. https://doi.org/10.1007/s00009-020-01605-4 doi: 10.1007/s00009-020-01605-4
![]() |
[45] |
Y. Bai, H. Kong, Existence of solutions for nonlinear Caputo-Hadamard fractional differential equations via the method of upper and lower solutions, J. Nonlinear Sci. Appl., 10 (2017), 5744–5752. http://dx.doi.org/10.22436/jnsa.010.11.12 doi: 10.22436/jnsa.010.11.12
![]() |
[46] | A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differenatial equations, North-Holland Mathematics Studies, Vol. 204, Amsterdam: Elsevier Science B.V., 2006. |
[47] | I. Podlubny, Fractional differential equations, New York: Academic Press, 1998. |
[48] |
R. Almeida, D. F. M. Torres, Computing Hadamard type operators of variable fractional order, Appl. Math. Comput., 257 (2015), 74–88. https://doi.org/10.1016/j.amc.2014.12.071 doi: 10.1016/j.amc.2014.12.071
![]() |
[49] |
O. Kahouli, D. Boucenna, A. Ben Makhlouf, Y. Alruwaily, Some new weakly singular integral inequalities with applications to differential equations in frame of tempered χ-fractional derivatives, Mathematics, 10 (2022), 3792. https://doi.org/10.3390/math10203792 doi: 10.3390/math10203792
![]() |
[50] |
A. Ben Makhlouf, D. Boucenna, A. M. Nagy, L. Mchiri, Some weakly singular integral inequalities and their applications to tempered fractional differential equations, J. Math., 2022 (2022), 1682942. https://doi.org/10.1155/2022/1682942 doi: 10.1155/2022/1682942
![]() |